6x+(1/2x-8)=3x-31

Solution for 6x+(1/2x-8)=3x-31 equation:

6x+(1/2x-8)=3x-31

We move all terms to the left:

6x+(1/2x-8)-(3x-31)=0


Domain of the equation: 2x-8)!=0

x∈R


We get rid of parentheses

6x+1/2x-3x-8+31=0

We multiply all the terms by the denominator


6x*2x-3x*2x-8*2x+31*2x+1=0

Wy multiply elements

12x^2-6x^2-16x+62x+1=0

We add all the numbers together, and all the variables

6x^2+46x+1=0

a = 6; b = 46; c = +1;
Δ = b2-4ac
Δ = 462-4·6·1
Δ = 2092
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{2092}=\sqrt{4*523}=\sqrt{4}*\sqrt{523}=2\sqrt{523}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(46)-2\sqrt{523}}{2*6}=\frac{-46-2\sqrt{523}}{12}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(46)+2\sqrt{523}}{2*6}=\frac{-46+2\sqrt{523}}{12}
$